Thiele Modulus Values Research Articles

Performance of mesoporous ceria supported Ru catalysts for oxidative degradation of aqueous solution of plastic monomer bisphenol A in a fixed bed flow reactor: Structure-activity insights, optimization of reaction conditions, kinetics, catalyst stability, and characterization

One of the important defects is the oxygen vacancy on metal oxide surfaces, and the defects function as the active sites in different catalytic reactions. Herein, Ru/CeO2-rod and Ru/CeO2-cube catalysts were prepared via the ESI method, and CeO2-rod and CeO2-cube supports were synthesized by hydrothermal processes. The activity of these catalysts was tested in a high-pressure fixed-bed reactor, and the oxidative degradation of industrial organic raffinate containing the plastic chemical bisphenol A (BPA) was compared with their oxygen vacancy concentration estimated after calcination. The Ru/CeO2-rod catalyst showed higher catalytic activity with 90 % BPA conversion at 453 K, 30 bar, 0.5 % Ru metal content, 15 O2 to BPA molar ratio, and 2 g−1.h−1 BPA feed weight hourly space velocity (WHSV). In contrast, 80 % BPA degradation was achieved using the Ru/CeO2-cube catalyst under similar reaction conditions. The higher oxygen vacancy concentration and the higher interfacial surface area between metal and the support of the CeO2-rod supported Ru catalyst triggered the higher BPA degradation at low oxidation conditions. Besides, the atomic ratio, Ce3+/Ce4+, is lower in the Ru/CeO2-rod catalyst than that of the Ru/CeO2-cube catalyst, indicating the higher Ce4+ available in the catalyst, leading to the higher degradation of BPA. Moreover, the Ru/CeO2-rod catalyst possesses {111} planes that boost the generation of energetic Ru to improve the pursuit of the catalyst. The Weisz-Prater modulus and Thiele modulus values indicated that the surface reaction is the rate limitation step and there is no diffusion limitation. TEM, HRTEM, XPS, Raman, H2-TPR, XRD, CO chemisorption and BET surface area analyzers were used to characterize the catalysts.

Generalized Linear Driving Force Formulas for Diffusion and Reaction in Porous Catalysts

Approximate models are a fast and most often precise tool for determining the effectiveness factor for heterogeneous catalysis processes that are realized in the real world. They are also frequently applied as robust transient models describing the work of a single catalyst pellet or as a part of a more complex model, for example, a reactor model, where mass balances for the gas phase and solid phase are necessary. So far, approximate models for diffusion and reaction processes have been presented for processes described by a single balance equation. In the present work, approximate models without the mentioned limitation are presented and discussed. In addition, simple rules are shown for the development of other complex approximate models without tedious derivation in the complex domain. The formulas considered in this work are typical long-time approximations of the transient process. The accuracy is good, especially in the range of small and intermediate Thiele modulus values.

Modeling and experimental study of hollow cylindrical catalysts for prediction of transient behaviors of suspension photoreaction systems by Langmuir-Hinshelwood kinetics

Effectiveness factor was derived for hollow cylindrical pellets with finite length by assuming Langmuir-Hinshelwood (LH) kinetics. Transient intra-particle concentration was predicted by solving reaction-diffusion equations using finite element method for various values of reaction parameters in LH kinetics and thickness of hollow core. At stationary state, numerical values of intra-particle concentration was integrated to calculate effectiveness factor (η) under various shape factors of hollow cylindrical pellet including infinite length as well as finite height. The results were compared with those of infinitely long cylinder and spherical pellets to determine optimum morphologies of catalytic pellets. η could be enhanced by increasing hollow-core size and decreasing aspect ratio of the pellet. For the same value of Thiele modulus, which was calculated from volume-equivalent radius, hollow cylindrical catalyst exhibited largest value of η, compared to solid cylindrical or spherical pellets. For simulation of catalytic reactors such as batch reactor, CSTR, and fixed bed, pseudo-steady state was adopted in material balance equation using computed values of η. The modeling results were compared with experimental data from batch photocatalytic reactor containing hollow catalytic fibers synthesized by electrospinning to determine reaction parameters in LH kinetics by minimizing errors of transient concentration between calculated and measured values using optimization method.

Modeling of Core-shell Catalytic Pellets of Inert Shell with Various Morphologies for Series Reactions

Modeling of core-shell catalytic pellets with inert shell was performed assuming three different particle shapes such as sphere, cylinder, and slab. For isothermal irreversible first order reaction, A→B, analytical solutions of intra-particle concentration of reactant were derived by solving reaction-diffusion equations, considering thickness of catalytically active core (r2), when the pellets were immersed in infinitely large medium. Unlike conventional pellets or core-shell pellets with inert core, effectiveness factor (η) was affected by r2 as well as ratio of effective diffusivity in inert shell and active core (Υ), and η could be enhanced by increasing Υ. η increased in the order of sphere > cylinder > slab due to surface area per unit volume of medium, and η increased with increasing Biot number (Bi), because of decreased external film resistance. For series reaction, A→B→C, transient change of the concentration of starting reactant (CA) and intermittent product (CB) in bulk fluid of reactor could be predicted by solving material balance equations assuming pseudo-steady state approximation and equal value of Thiele modulus for both reactions in batch reactor and CSTR. When B is desirable product, CB increased by increasing Υ and decreasing r2. Similar to conventional pellets, CB in batch reactor decreased by increase of distribution coefficient of B on surface of pellet, KB, due to increased adsorption, and increased in the order of sphere < cylinder < slab, similar to conventional pellets. In CSTR, CB at steady state could be enhanced by decreasing amount of catalyst and Thiele modulus (Φ), implying that performance of reactor can be controlled by adjusting various reaction parameters. From numerical solutions of reaction diffusion equations for series reactions, the effect of r2 and Υ were also confirmed from transient value of CA and CB in exit stream calculated by finite element method.

Theoretical Analysis of Mass Transfer Behavior in Fixed-Bed Electrochemical Reactors: Akbari-Ganji’s Method

The theoretical model for a packed porous catalytic particle of the slab, cylindrical, and spherical geometries shape in fixed-bed electrochemical reactors is discussed. These particles have internal mass concentration and temperature gradients in endothermic or exothermic reactions. The model is based on a nonlinear reaction–diffusion equation containing a nonlinear term with an exponential relationship between intrinsic reaction rate and temperature. The porous catalyst particle’s concentration is obtained by solving the nonlinear equation using Akbari-Ganji’s method. A simple and closed-form analytical expression of the effectiveness factor for slab, cylindrical, and spherical geometries was also reported for all values of Thiele modulus, activation energy, and heat reaction. The accordance with results of a reliable numerical method shows the good accuracy that their approximate solution yields.

Assessment of effectiveness factor in porous catalysts under non-symmetric external conditions of concentration

Abstract the present paper, the effectiveness factor of porous catalytic particles is evaluated in the absence of boundary conditions symmetry over the external surface by computational fluid dynamic (CFD) techniques. The first-order kinetics of decane oxidation, already evaluated experimentally, is taken as a representative reaction. Our study arises from the fact that, in the open literature, the effectiveness factor is usually calculated considering conditions of symmetry of concentration field around particles. However, depending on the fluid dynamics of the system, such conditions are not always established and, thus, our work aims at studying for the first time the behaviour of particle catalysts with non-uniform concentration fields over the surface. In particular, the effectiveness factor of the particles in a catalytic layer is calculated in the absence of symmetry by changing several parameters (temperature, tortuosity and mean pore diameter of particle) using two different methods, named Sphere-by-Sphere (SbS) and Equisized-Volume (EV), respectively. The results of these two methods are then compared to the theoretical one obtained in the presence of spherical symmetry. As a main result, we found that, for moderately low values of Thiele modulus (&lt;1.3 ca.), the analytical expression of the effectiveness factor obtained under spherical symmetry can be also applied in non-symmetric conditions. On the contrary, this cannot be done for higher values of Thiele modulus, for which we propose an empirical correlation of the effectiveness factor based on a corrected Thiele modulus. The efficacy of our approach is stated by the fact that pseudo-homogeneous-mode simulations of the heterogeneous system show results that match very well those obtained in heterogeneous mode, with an important reduction of calculation time and memory. The presented methodology can be also applied to n-order kinetics.

Modelling reaction and diffusion in a wax-filled hollow cylindrical pellet of Fischer Tropsch catalyst

Previous modelling of fixed-bed, Fischer-Tropsch (FT) reactors has demonstrated the advantages relative to spherical pellets of using cylindrical and shaped pellets to provide improved transport attributes under conditions relevant to industrial operation. However, mass transport models have focussed on the investigation of transport within pellets with spherical symmetry, whilst detailed investigations of more complex shapes have not been undertaken. Here, a pseudo-isothermal, steady-state, two-dimensional model was investigated for catalyst pellets of cylindrical form, both solid and hollow. A cobalt-based catalyst was considered at conditions where the rate of condensable hydrocarbon generation is large enough to result in the accumulation of liquid hydrocarbons in the pores of a catalyst. It was found that effectiveness factors were bounded by those of sphere and slab above and below Thiele moduli of ∼0.75 and ∼1.15, respectively, for the conditions examined, with the effectiveness factors exceeding those of both sphere and slab models between these moduli. Here, comparisons were made on the basis of the characteristic diffusion length, the catalyst particle’s volume divided by its external surface area. However, values of the FT chain growth parameter, α, between these values of Thiele modulus were lower than those of both sphere and slab geometry, and thus under these conditions hollow cylinders gave the greatest methane selectivity.

Diffusion–reaction phenomenon for negative-order reactions in flow reactors

The phenomenon of simultaneous diffusion, reaction, and external mass transfer in a spherical catalyst particle is studied for a reaction exhibiting negative order with respect to the reactant species. With external mass transfer considered, it is shown that for a reaction exhibiting a negative order for all the concentrations, a solution to the diffusion–reaction equations does not exist for high values of Thiele modulus. An algorithm is developed to calculate the concentration profile in a spherical catalyst particle for a negative-order reaction. With the proposed algorithm, the solution is obtained even for conditions for which a solution was not found in previous studies. The algorithm successfully captures the multiplicity of the solutions as well as the transition from kinetic/internal-diffusion controlled regime to the external mass-transfer controlled regime. The developed algorithm is further utilized to calculate the axial concentration profile in a packed-bed reactor for a negative-order reaction. Additionally, the radial concentration profile in a spherical particle is evaluated for a multi-component reaction system wherein the reaction exhibits both positive and negative order with respect to various species. The algorithm is particularly useful for power-law type negative order kinetics which are widely reported in the literature.

Approximating Catalyst Effectiveness Factors with Reaction Rate Profiles

A novel approximate solution for catalyst effectiveness factors is presented. It is based on carefully selected approximate reaction rate profiles, instead of typical assumption of composition profiles inside the catalyst. This formulation allows analytical solution of the approximate model, leading to a very simple iterative solution for effectiveness factor for general nonlinear reaction stoichiometry and arbitrary catalyst particle shape. The same model can be used with all practical Thiele modulus values, including multicomponent systems with inert compounds. Furthermore, the correct formulation of the underlying physical model equation is discussed. It is shown that an incorrect but often-used model formulation where convective mass transfer has been neglected may lead to much higher errors than the present approximation. Even with a correctly formulated physical model, rigorous discretization of the catalyst particle volume may have unexpectedly high numerical errors, even exceeding those with the present approximate solution. The proposed approximate solution was tested with a number of examples. The first was an equimolar reaction with first order kinetics, for which analytical solutions are available for the standard catalyst particle geometries (slab, long cylinder, and sphere). Then, the method was tested with a second order reaction in three cases: 1) with one pure reactant, 2) with inert present, and 3) with two reactants and non-stoichiometric surface concentrations. Finally, the method was tested with an industrially relevant catalytic toluene hydrogenation including Maxwell-Stefan formulation for the diffusion fluxes. In all the tested systems, the results were practically identical when compared to the analytical solutions or rigorous finite volume solution of the same problem.

Synthesis of Hierarchically Porous H[P, Al]‐ZSM‐5 and its Catalytic Performance in Coupling Transformation of Methanol with 1‐Butene

AbstractHierarchically porous P substituted HZSM‐5 (H[P, Al]‐ZSM‐5‐AT‐0.05 M ) was successfully synthesized by isomorphous substitution method with introduced P atoms into ZSM‐5 zeolite frameworks followed by alkali treatment. H[P, Al]‐ZSM‐5‐AT‐0.05 M was characterized by XRD, ICP‐OES, SEM, FT‐IR, N2 adsorption‐desorption, NH3‐TPD, XPS, Py‐IR, and TG‐DTA, respectively. The surface area, pore volume, particle size and P/Al (molar ratio) are 328 m2.g−1, 0.23 cm3.g−1, 563 nm and 0.027, respectively. The catalytic performance of H[P, Al]‐ZSM‐5‐AT‐0.05 M in coupling reaction of methanol with 1‐butene was investigated in the fixed‐bed reactor. On the condition of reaction temperature at 500°C under atmosphere pressure, the main product is propylene with the selectivity and yield of 33.7% and 31.0%, respectively. Hydrogen transfer index was calculated to 0.24. Moreover, its diffusion limitation was investigated with the self‐etherification of benzyl alcohol reaction. The value of Thiele modulus ( ) and effectiveness factor ( ) are 0.08 and 16.9, respectively.

Pore diffusion effects on catalyst effectiveness and selectivity of cobalt based Fischer-Tropsch catalyst

Abstract In this study we investigate performance characteristics (catalyst effectiveness, CH4 selectivity, and hydrocarbon product distribution) of with a highly active Co/Re/Al2O3 catalyst particle for Fischer-Tropsch synthesis. In numerical simulations we utilize kinetic parameters for CO consumption rate, CH4 formation rate and hydrocarbon formation rates (C2+ hydrocarbons) determined from experiments with this catalyst to study effects of catalyst activity, catalyst particle shape (sphere, slab, solid and hollow cylinder), size (i.e. diffusion length), catalyst distribution (uniform vs. eggshell type distribution for a spherical particle) and process conditions (temperature, pressure, syngas composition and conversion level) on the catalyst performance. With increase in Thiele modulus (i.e. particle size at a fixed set of process conditions) we observe increasing H2/CO ratio profile towards the center of the particle resulting in increase of local and average CH4 selectivity. The goal is to find conditions which allow one to use sufficiently large particles to reduce pressure drop, while avoiding negative influence of diffusional limitations on selectivity and activity. For each catalyst particle shape we determined values of Thiele modulus, i.e. characteristic length of diffusion, corresponding to the upper limit of the kinetic region, and investigated how it changes with operating conditions. We found that simultaneous increase of pressure and the use of syngas with H2/CO feed ratio of 1.4–1.7 is the best strategy for mitigating the negative impact of intraparticle diffusional limitations on CH4 selectivity. For a spherical particle of 1 mm in diameter, one can achieve CH4 selectivity of 5.6% with catalyst effectiveness factor of 1.07 at the reactor inlet by operating at 50 bar, 473 K and H2/CO = 1.4.

Combustion of spherically shaped large wood char particles

This paper examines the effects of temperature and oxygen enrichment on conversion of large wood char particle under non-isothermal conditions. Combustion air and its combination with oxygen are examined at temperatures of 800 and 850°C for identical initial char weight, diameter and reaction time. One-film ash segregated core and random pore models are used to characterize the combustion condition. The combustion regimes for these particles are found to be near kinetic-diffusion controlled based on the ash segregated core model. Diffusion and reaction rate relationships are also characterized. Oxygen enriched conditions provided greater reactivity as compared to non-enriched oxidation conditions. For the ash segregated core model, the lowest and highest estimated activation energy values are 123kJ/mol and 180kJ/mol, respectively. The activation energies obtained with the random pore model showed inconsistencies. Thiele modulus values varied from 500 to 1000, indicating that the surface reaction rates are significantly faster than pore diffusion rates with oxygen being mostly consumed at the particle surface.

A Theoretical Model of pH-Based Potentiometric Biosensor Based on Immobilized Enzyme Membrane

A theoretical model for the non steady-state response of a pH-based potentiometric biosensor immobilizing organophosphorus hydrolase (OPH) is discussed. The model is based on a system of five coupled nonlinear reaction-diffusion equations under non steady-state conditions for enzyme reactions occurring in potentiometric biosensor that describes the concentration of substrate and hydrolysis products within the membrane. New approximate analytical expressions for the concentration of the substrate (organophosphorus pesticides (OPs)) and products are derived for all values of Thiele modulus and buffer concentration using new approach of homotopy perturbation method. The analytical results are also compared with numerical ones and a good agreement is obtained. The obtained results are valid for the whole solution domain.

An Effectiveness Model for Electrochemical Reactions in Electrodes of Intermediate-Temperature Solid Oxide Fuel Cells

A new effectiveness model is proposed for efficiently predicting the electrochemical performance of electrodes in intermediate-temperature solid oxide fuel cells (IT-SOFCs). An electrochemical reaction/charge transport problem is formulated based on the assumptions that the fuel cell processes in the active functional layers occur under uniform operating conditions and that the transfer current-overpotential relation follows the symmetric Butler-Volmer equation. The electrochemical effectiveness is calculated for various Thiele modulus values, and a simple correlation is developed for retrieving these effectiveness factor data. Finally, the validity of the effectiveness model is demonstrated by comparing the results with those obtained from the detailed electrode microscale model.

Multiplicity results and closed-form solution for catalytic reaction in a flat particle

Assume a flat geometry for the particle and that conductive heat transfer is negligible compared to convective heat transfer in the study of heat and mass transfer for a catalytic reaction within a porous catalyst flat particle. This article shows the governing differential equation, which is the direct result of a material and energy balance, is exactly solvable and furthermore, gives exact analytical solution in the implicit form for further physical interpretation. A full discussion is given and, it is also revealed that the problem may admit unique, dual or even more triple solutions depending on the values of Thiele modulus and other parameters of the model.

Reduction of ethylenediaminetetraacetic acid iron(III) by Klebsiella sp. FD-3 immobilized on iron(II, III) oxide poly (styrene-glycidyl methacrylate) magnetic porous microspheres: Effects of inorganic compounds and kinetic study of effective diffusion in porous media

Fe3O4 poly (styrene-glycidyl methacrylate) magnetic porous microspheres (MPPMs) were introduced to immobilize Klebsiella sp. FD-3, an iron-reducing bacterium applied to reduce Fe(III)EDTA. The effects of potential inhibitors (S2−, SO32−, NO3−, NO2− and Fe(II)EDTA-NO) on Fe(III)EDTA reduction were investigated. S2− reacted with Fe(III)EDTA as an electron-shuttling compound and enhanced the reduction. But Fe(III)EDTA reduction was inhibited by SO32− and Fe(II)EDTA-NO due to their toxic to microorganisms. Low concentrations of NO3− and NO2− accelerated Fe(III)EDTA reduction, but high concentrations inhibited the reduction, whether by free or immobilized FD-3. The immobilized FD-3 performed better than freely-suspended style. The substrate mass transfer and diffusion kinetics in the porous microspheres were calculated. The value of Thiele modulus and effectiveness factors showed that the intraparticle diffusion was fairly small and neglected in this carrier. Fe(III)EDTA reduction fitted first-order model at low Fe(III)EDTA concentration, and changed to zero-order model at high concentrations.

Study of intraparticle diffusion-reaction of substrate for Michaelis–Menten kinetics in a porous slab catalyst

The effect of internal diffusion on the overall reaction rate in a biocatalyst having slab geometry containing an immobilized enzyme or cells have been investigated theoretically. Zero-order, first-order and Michaelis-Menten kinetics were studied. The exact solutions for zero-order and first- order reactions were studied to verify our numerical algorithm which was later used to obtain solution for the Michaelis-Menten kinetic. The concentration profiles within the catalyst slab were obtained as a function of Thiele modulus which in turn were used to evaluate effectiveness factor. The exact solutions for zero and first order reactions can be obtained analytically. However one has to resort to numerical solution for Michaelis Menten kinetics as the resulting nonlinear differential equation cannot be solved analytically for the exact solution. Thus, for the Michaelis–Menten kinetics, the diffusion-reaction equation is solved using numerical method employing an explicit finite difference scheme which proved to be stable and accurate. A simple third order polynomial solution to the differential equation is also proposed. The approximate solution shows close agreement (error about less than 10%) with the numerical solution within the range of parameters of practical significance such as Thiele modulus values up to 8. Thus the approximate solution obtained in this work gives quite satisfactory results for a wide range of Thiele modulus compared to that reported in the literature. The nutrients diffuse deeper into the pellet with decreasing Thiele moduli for the three rate kinetics studied. The effectiveness factor decreases with increasing Thiele moduli which is in agreement with the trend in concentration profile for all the cases investigated and the range of parameters studied. Keywords: Diffusion-reaction; Michaelis–Menten kinetics; Immobilized slab biocatalyst; Finite difference method; Approximate solution

Continuum heterogeneous biofilm model—A simple and accurate method for effectiveness factor determination

We present a novel analytical approach to describe biofilm processes considering continuum variation of both biofilm density and substrate effective diffusivity. A simple perturbation and matching technique was used to quantify biofilm activity using the steady-state diffusion-reaction equation with continuum variable substrate effective diffusivity and biofilm density, along the coordinate normal to the biofilm surface. The procedure allows prediction of an effectiveness factor, η, defined as the ratio between the observed rate of substrate utilization (reaction rate with diffusion resistance) and the rate of substrate utilization without diffusion limitation. Main assumptions are that (i) the biofilm is a continuum, (ii) substrate is transferred by diffusion only and is consumed only by microorganisms at a rate according to Monod kinetics, (iii) biofilm density and substrate effective diffusivity change in the x direction, (iv) the substrate concentration above the biofilm surface is known, and (v) the substratum is impermeable. With this approach one can evaluate, in a fast and efficient way, the effect of different parameters that characterize a heterogeneous biofilm and the kinetics of the rate of substrate consumption on the behavior of the biological system. Based on a comparison of η profiles the activity of a homogeneous biofilm could be as much as 47.8% higher than that of a heterogeneous biofilm, under the given conditions. A comparison of η values estimated for first order kinetics and η values obtained by numerical techniques showed a maximum deviation of 1.75% in a narrow range of modified Thiele modulus values. When external mass transfer resistance, is also considered, a global effectiveness factor, η(0) , can be calculated. The main advantage of the approach lies in the analytical expression for the calculation of the intrinsic effectiveness factor η and its implementation in a computer program. For the test cases studied convergence was achieved quickly after four or five iterations. Therefore, the simulation and scale-up of heterogeneous biofilm reactors can be easily carried out.

강인한 바이오필터설계를 위한 바이오필터모델: 2. 동적 바이오필터모델

A dynamic biofilter model was suggested to integrate the effect of biofilter-medium adsorption capacity on the removal efficiency of volatile organic compound (VOC) contained in waste air. In particular, the suggested biofilter model is composed of four components such as biofilm, gas phase, sorption volume and adsorption phase and is capable of predicting the unsteady behavior of biofilter-operation. The process-lumping model previously suggested was limited in the application for the treatment of waste air since it was derived under the assumption that the adsorbed amount of VOC equilibrated with biofilter-media would be proportional to the concentration of dissolved VOC in the sorption volume of biofilter-media. Therefore a Freundlich adsorption isotherm was integrated into a robust biofilter process-lumping model applicable to a wide range of VOC concentration. The values of model parameters related to biofilter-medium adsorption were obtained from the dynamic adsorption column experiments in the preceding article and literature survey. Furthermore a separate biofilter experiment was conducted to treat waste air containing ethanol and the experimental result was compared with the model predictions with various values of Thiele modulus (). The obtained value of Thiele modulus () was close to 0.03.

Dirichlet problem for convection–diffusion–reaction inside a permeable cylindrical porous pellet

The present article deals with the study of convection and diffusion coupled with either zero or first order reaction inside a permeable circular cylindrical porous pellet under oscillatory flow. Unsteady Stokes equations are used for the flow outside the permeable porous pellet and Darcy’s law is used inside the pellet. We use the stream function approach in order to solve the hydrodynamic problem. Then the convection–diffusion–reaction problem is formulated and solved analytically for both zero order and first order rate of nutrient uptake. The Dirichlet boundary condition, which can be achieved by neglecting the external mass transfer resistance, is used at the surface of permeable porous pellet. Also in case of zero order, an optimality criterion, which is a relationship between the Peclet number and the Thiele modulus, is proposed to avoid the starvation everywhere inside the pellet. Based on this criterion, classification is done in order to identify the regions of nutrient sufficiency and starvation. A comparison is also made with nutrient transport inside a spherical porous pellet. It is observed that in case of zero order, for a fixed combination of other parameters, spherical pellet demands a higher value of Thiele modulus compared to the cylindrical pellet in order to force starvation. Moreover, in case of first order reaction, one does not witness starvation zones either in cylindrical pellet or in spherical pellet.